Why didn't the Mitzryim magic makers just take away the blood from the first makkah to prove to Moshe that he was just doing magic and Hashem had nothing to do with it: why did they add blood (Exodus 7:22)?

Not from jewish source, but I bet those magician simply use sleight of hand. There is indeed a snake that looks like sticks if handled properly. It's done by grabbing the snake's neck. In a sense, it's Moses that's mimic those egyptian magician except that he mimics it with real "magic". Perhaps there is a trick to turn water into blood (putting dye). The other way around is beyond their skills.
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Jim ThioDec 22 '12 at 11:30

Hence the story that Moses grab his big snake by the tail is actually significant. It's like David Coperfield waving leather on top of his head to show he uses no strings. Moses effectively say, look, I used a big species of snake that do not use your cheap acupuncture trick.
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Jim ThioDec 22 '12 at 11:32

The Torah clearly states that all water throughout the land was turned into blood. For the magicians to also turn water into blood, they must have first turned the blood back into water.
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Raffy Van der VaartJan 9 '13 at 16:32

Rav Hirsch suggests an alternate explanation of the magicians’ behavior according to your suggestion: that they were attempting to undo the effects of the plague with no success—or in the case of the frogs, more frogs came when they attempted to banish them. After their third failure, they acknowledged that it was “God’s Finger” at work.

Imagine there was a magician claiming to have a super-natural ability to turn water into blood, and you want to discredit him and prove that it's just a trick. You would need to perform the exact illusion that the magician was performing, turn water into blood. Doing the reverse would not discredit the initial "miracle" that the magician performed.

Similarly, the Egyptian magicians wanted to show Pharaoh that what Hashem could do wasn't special. In order to do that the magicians needed to repeat the same exact effect that Moshe performed. Turning it back into water would not discredit Hashem's powerful ability to turn it to blood.